US8406280B2 - System and method for mitigating severe multipath interference for geolocation and navigation - Google Patents
System and method for mitigating severe multipath interference for geolocation and navigation Download PDFInfo
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- US8406280B2 US8406280B2 US12/406,456 US40645609A US8406280B2 US 8406280 B2 US8406280 B2 US 8406280B2 US 40645609 A US40645609 A US 40645609A US 8406280 B2 US8406280 B2 US 8406280B2
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S5/00—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations
- G01S5/02—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations using radio waves
- G01S5/0273—Position-fixing by co-ordinating two or more direction or position line determinations; Position-fixing by co-ordinating two or more distance determinations using radio waves using multipath or indirect path propagation signals in position determination
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
- G01S19/01—Satellite radio beacon positioning systems transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
- G01S19/13—Receivers
- G01S19/22—Multipath-related issues
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
- G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
- G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
- G01S19/42—Determining position
- G01S19/45—Determining position by combining measurements of signals from the satellite radio beacon positioning system with a supplementary measurement
- G01S19/47—Determining position by combining measurements of signals from the satellite radio beacon positioning system with a supplementary measurement the supplementary measurement being an inertial measurement, e.g. tightly coupled inertial
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B1/00—Details of transmission systems, not covered by a single one of groups H04B3/00 - H04B13/00; Details of transmission systems not characterised by the medium used for transmission
- H04B1/69—Spread spectrum techniques
- H04B1/707—Spread spectrum techniques using direct sequence modulation
- H04B1/7097—Interference-related aspects
- H04B1/711—Interference-related aspects the interference being multi-path interference
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B2201/00—Indexing scheme relating to details of transmission systems not covered by a single group of H04B3/00 - H04B13/00
- H04B2201/69—Orthogonal indexing scheme relating to spread spectrum techniques in general
- H04B2201/707—Orthogonal indexing scheme relating to spread spectrum techniques in general relating to direct sequence modulation
- H04B2201/70715—Orthogonal indexing scheme relating to spread spectrum techniques in general relating to direct sequence modulation with application-specific features
Definitions
- GNSS Global Navigation Satellite Systems
- GPS Global Positioning System
- GLONASS Russian GLONASS
- LOS clear line of sight
- GNSS Global System for Mobile Communications
- APS Assisted GPS
- UTC Ultra-Tight Coupling
- NLOS non-line of sight
- Multipath signals always arrive “late” compared to the LOS signal, thus creating an error in the measured range and corrupting the user position estimate. This problem is especially acute indoors and in urban canyons, due to multiple reflection-generating objects which surround a user (building walls, furniture, cars, etc.).
- MEDLL Multipath Estimation Delay Lock Loop
- a MEDLL receiver has many correlators which integrate the satellite signal at different delays (compared to typically three correlators for a traditional receiver) against the known code of the transmitted signal.
- the result is a profile of a GNSS satellite signal's correlation with the code replica, sampled at an array of points.
- LOS and all multipath components contribute to this profile, forming a complex signature.
- the MEDLL can discriminate the individual signal components, thus can be used to isolate LOS from multipath. This method is effective if there are few dominant signal components, and if one of the dominant components is the LOS signal. If there are many multipath components and/or if LOS is not present or weak, the MEDLL receiver does not yield a reliable LOS measurement.
- the present disclosure presents novel and advantageous systems and methods for discriminating between LOS and NLOS signal paths in a radio frequency (RF) receiver such as a GNSS navigation receiver.
- RF radio frequency
- the present disclosure provides for a method referred to as Synthetic Aperture Line of Sight Assessment, or SALSA.
- SALSA method uses the direction of arrival for a signal to discriminate between LOS and NLOS and is especially beneficial in environments where the amplitude of the multipath signals exceeds that of the LOS signal, and where the multipath delays are within the inverse of the signal bandwidth.
- the present disclosure provides for a method referred to as Genetic Algorithm for Multipath Elimination, or GAME.
- GAME Genetic Algorithm for Multipath Elimination
- the GAME method identifies and isolates signals that have been incorrectly identified as LOS signals. For a user navigating in urban or indoor environment, some LOS signals may be completely blocked by building walls, etc. Thus, there is no guarantee that a pseudorange measurement is not corrupted by the multipath, even if the most sophisticated algorithms and hardware is used for processing each signal. Therefore, one needs to have a method to identify LOS paths among received signals. If there are multiple signals received, then it is possible to check the consistency of several measurements to find if any of them are corrupted.
- RAIM Receiveiver Autonomous Integrity Monitoring
- GAME GAME is designed to serve this purpose.
- the SALSA and GAME methods are complementary and are designed to extract a LOS measurement in a multipath environment if the LOS signal is present and if the user is moving, or if a signal is received over time. SALSA and GAME may also be used to support Direction Finding algorithms for emitter geolocation.
- WAFFLES Weighted Average Functionality For Limiting Error Sources
- WAFFLES uses linear combinations of TDOA measurements in such way, that it largely cancels effects of timing, calibration, and geolocation errors.
- An alternative and complementary navigation method to GPS means is the use of various terrestrial signals rather than GPS signals. For example, signals for digital TV, from cell phone base stations, and Wi-Fi stations can be used to navigate in the areas where such signals are available. Since these signals are not designed for navigation, they lack several important navigation features, most importantly, accurate timing. This makes it necessary to use reference stations to timestamp signal features. Information about timestamps can be transmitted to the user via a communication channel.
- TDOA Time Division Multiple Access
- location of signal sources is not always known with high accuracy. For example, location of a cell base tower may be obtained from original engineering drawings, but there is always a possibility that location data will have some inaccuracies. Errors in locating signal sources directly impact user navigation accuracy.
- Another problem deals with calibration and timing errors at the reference stations. TDOA at the user assumes that timing measurements at the reference stations are accurate. This may be not the case if calibration and synchronization of reference stations is inaccurate.
- TDOA measurements at the user will be impacted by timing and calibration errors in two ways: (1) directly, and (2) due to errors in source geolocation.
- WAFFLES addresses this problem by using linear combinations of TDOA measurements in such way, that it largely cancels effects of timing, calibration, and geolocation errors.
- the present disclosure provides for a method of navigation in an environment where the navigation signals may be obstructed, referred to as LEAF.
- RF navigation under foliage presents its own set of challenges. The same may be true for some other complex environments, where signal propagation to the user is subject to diffused scattering. The earliest arrival of the signal is due to the direct signal component, and its measurement is the goal of the pseudorange estimate by the receiver. However, this component arrives at the receiver along with all scattered signal components, which cannot be separated from the direct component and which corrupt the pseudorange measurement.
- the LEAF method is designed to estimate pseudorange using statistical assumptions about the propagation medium. Individual scatterers (e.g., leaves and twigs in the foliage) produce random phases and amplitudes of the signal at different delays; however statistical properties of the received signal are predictable and can be used. The method finds the most likely signal delay from correlation measurements obtained by an array of correlators.
- FIG. 1 is a simplified pictorial representation of a multipath signal environment.
- FIG. 2 is a simplified pictorial representation of signal correlations in a multipath environment.
- FIG. 3 is a simplified flow diagram of one embodiment of the Synthetic Aperture Line of Sight Assessment (SALSA) method.
- SALSA Synthetic Aperture Line of Sight Assessment
- FIG. 4 is a simplified pictorial representation of the Genetic Algorithm for Multipath Elimination (GAME) algorithm.
- FIG. 5 is a simplified pictorial representation of the effects of timing and calibration errors on source geolocation.
- FIG. 6 is a simplified pictorial representation of the Weighted Average Functionality For Limiting Error Sources (WAFFLES) algorithm.
- FIG. 7 is a simplified pictorial representation of the LEAF algorithm.
- Determining the Direction of Arrival (DOA) of a signal is a useful method to discriminate LOS from multipath.
- the LOS signal generally comes from the direction of the transmitting satellite, while the multipath signals come from the direction of the reflector.
- the DOA discriminator is not constrained by assumptions of large delay and relatively weak multipath.
- Traditional methods for separating signals by DOA require an antenna array. Of course, it is not always practical for a user to carry an antenna array. In the mobile receiver environment, it is desirable to use a small, one-element omni-directional antenna.
- the present disclosure describes a method for using an omni-directional antenna for DOA by utilizing an antenna array processing effect created by exploiting user motion to synthesize the array aperture.
- a walking user 100 moves at the speed of approximately 2 m/s beginning at time t 0 .
- user 100 may receive a multipath signal 110 and a LOS signal 120 from GNSS transmitting satellite 130 .
- the user position has changed by many wavelengths (e.g., wavelength of the GPS L 1 frequency is 19 cm), and may receive a multipath signal 140 and a LOS signal 150 from satellite 130 .
- a receiver will process the received signals at times t 0 and t 1 (and at additional times t n ) to separate LOS from multipath and, in effect, create a large synthetic aperture array. In the case of a user in a vehicle traveling at an increased speed, the resulting synthetic aperture array will be larger.
- a GNSS receiver can correlate an incoming satellite signal with a known code replica to determine complex (I and Q) correlations. Each correlation requires integrating over some period of time, ⁇ t. This time interval is referred to in this disclosure as a sub epoch.
- a bank of N correlators each having an equally spaced code delay, produces N correlation measurements for each sub epoch.
- the receiver accumulates correlation measurements from multiple sub epochs over each epoch T.
- there is a two-dimensional array of correlations where one dimension is the code delay, and another dimension is time (sub epoch).
- FIG. 2 is a simplified pictorial representation of signal correlations in a multipath environment.
- Signal correlations are measured in two dimensions: at different time instances (sub-epochs) 200 as well as at different delays of signal replica 210 .
- the top portion of the figure illustrates a two-dimensional matrix of resulting complex correlations, each represented with an arrow 220 .
- a spectral analysis algorithm 230 transforms the time domain into the frequency domain 240 .
- signals reaching the receiver via different paths become separated, and each path exhibits a maximum correlation at a particular delay, shown as a tip of a triangle 250 .
- the earliest arrival is likely to be the line-of-sight.
- phase rate bins For each phase rate bin, there may still be some correlation profile in the replica delay domain that may be due to relatively few paths which have the corresponding phase rate (e.g., arrive from approximately the same corresponding direction). However, only a few paths will contribute to a particular phase rate bin, and therefore these paths can be resolved by some conventional techniques, such as MEDLL. The result of this procedure will be multiple, separately identified paths, each with its own phase rate and delay. The earliest path (smallest delay) is a candidate for the LOS.
- a user may not move with a constant velocity and along a straight line.
- the disclosed methods can be applied to any trajectory if the LOS components are added coherently, and multipath components are added destructively. This can be done if the relative trajectory is approximately known during the epoch, which generally requires using an inertial measuring unit (IMU).
- IMU inertial measuring unit
- the correlation phases for each correlation measurement may be rotated in such a way that LOS components should have the same phase. This step will be referred to as motion compensation.
- LOS corresponds to the zero phase rate (user “standing still” in the framework tied to the user) and the Fourier transform at the zero phase rate is the same as integration over time. This example, though not an optimal implementation, shows that SALSA has some similarities with integration.
- SALSA accounts for imperfections in motion compensation.
- IMU inertial measuring unit
- IMU inertial measuring unit
- perfect motion compensation would be possible, and the LOS component is guaranteed to be in the zero phase rate bin.
- integration would achieve the same result as the Fourier transform.
- the estimate of the user trajectory is imperfect, and the user clock has a drift.
- the true signature of the LOS may appear not in the zero phase rate bin, but somewhere in its vicinity, and a Fourier transform may be used to find it.
- the position of the LOS-induced maximum in the phase rate domain is an indication of the errors in the clock drift and trajectory estimate. In essence, it is the phase rate residual. When multiplied by the epoch duration, it becomes the residual in ADR (accumulated delta range), which is a valuable measurement in itself, commonly used by many navigation systems.
- ADR accumulated delta range
- SALSA assists in isolating the LOS component from multipath.
- Some multipath components may be located in phase rate bins which are poorly separated from the location of the LOS bin. Each component will show up as a maximum in the phase rate (Fourier) domain. If maxima are not well separated, a tail of one maximum may contaminate a measurement for another maximum. A plain time-integration ignores this effect, and suffers from multipath correlations “leaking into” the integration result.
- SALSA uses windowing (e.g., Kaiser-Bessel window) to improve path separation.
- SALSA may be used with beamforming and direction finding to improve isolation of the LOS signal.
- Each path contributes a spectral peak in the phase rate (Fourier) domain.
- DF beamforming and direction-finding
- examples include MUSIC, Maximum Entropy Methods, and weighted subspace fitting.
- SALSA uses a modified version of the MUSIC algorithm to analyze correlations across RF channels and delays, and thus does not require user motion.
- a matrix C* can be formed
- a ( z ) 1 +c 1 z ⁇ 1 + . . . +c n z ⁇ n .
- Roots of that polynomial for z ⁇ 1 happen to correspond to components of the signal:
- Equations [12] for different l form a linear system with linearly independent coefficients a k e i ⁇ k l . Thus, they can only be satisfied if
- Equation [14] must hold for any k, which means that b k are simply ⁇ k ⁇ 1 (not necessarily in the same order).
- a receiver receives and digitizes an RF signal 300 .
- the following sub-steps are performed:
- the received signal 300 is correlated with a replica waveform using a bank of correlators 310 with the following characteristics:
- the 2D correlation matrix is corrected for the user motion 330 with the following process:
- a spectrum estimation algorithm 340 is applied in the time domain to the correlation matrix to separate out the different signal paths by direction of arrival (shown in FIG. 2 as “FFT over sub-epochs”).
- Example spectrum estimation algorithms include windowing and Fourier transform, principal component analysis, singular spectrum analysis, maximum entropy method, MUSIC method, and Modified MUSIC method.
- An example pictorial of the resultant Replica Delay-Phase rate plot, along with the observation of the earliest arriving signal path is shown in FIG. 2 .
- spectral estimation algorithms may also be applied to frequency and channel domains for additional multipath mitigation.
- the LOS path is selected 350 using one or more of the following criteria:
- the PR and Doppler measurements are used to estimate user position using standard navigation techniques such as Kalman filtering 370 .
- High-rate IMU data 380 can be used to augment the measurements in the navigation filter 370 .
- the SALSA MUSIC method is based on two observations. First, isolating LOS and multipaths in a multi-channel signal (such as OFDM) by a moving user is mathematically equivalent to the problem of spectral estimation in two dimensions. The first dimension is the time domain, and the second dimension is the channel domain. Second, the conventional MUSIC algorithm does not work well for the problem at hand, because LOS and multipath signals are highly correlated (basically, fully correlated). However, the modified MUSIC algorithm works.
- the SALSA MUSIC method can be summarized as follows:
- Each frequency channel is tracked and demodulated individually.
- the replica signal for each channel is correlated with the incoming signal to obtain I and Q measurements (complex correlations). This creates a 2D matrix of data (number of sub-epochs ⁇ number of frequency channels complex values).
- a reference station supplies its own measurements of the complex amplitude for each channel.
- the user rotates/scales measurements to compensate for any clock drifts at the signal source and for any variations in amplitude (e.g., due to power control at the signal source).
- the algorithm corrects for the user motion, which means that phases of measurements are rotated.
- the Modified MUSIC algorithm is applied to one channel (the middle one) in the 2D matrix of the data. This algorithm determines the spectrum (which can be mapped to the directions of arrival for different paths in the synthetic aperture array). The algorithm finds a fixed number of paths, regardless of the true number of paths in the data. For example, it may look for N-10 paths. If the RF propagation channel contain fewer than 10 paths, the algorithm will find spurious paths (described below); if the RF propagation channel contains more than 10 paths, some weaker ones will not be found.
- the next task is to separate the seeds from the chaff, i.e. to eliminate spurious paths.
- the reasoning goes like this: if a beam is formed on the spurious path and all other paths are nulled (which includes all real paths) then the resulting amplitude of the signal will be relatively small. This applies to both stages of the beamforming. Thus, 30 path combinations are sorted by their amplitudes, and some in the bottom of the list are cut off.
- the selection is based on three criteria:
- inertial measuring units (IMU) and UTC can be used to get multiple measurements for a single epoch.
- Multiple correlation measurements can be collected over one epoch, e.g. 100 measurements, separated by 10 ms are collected over one second. Rather than integrating or averaging these measurements (as it is done by techniques for long integration of GPS signals), measurements are processed as a set.
- measurements are corrected for the user motion, in such way that for the LOS path the phase of the signal remains constant or changes linearly.
- array processing techniques are used to isolate paths.
- Array processing techniques are applied to the set of measurements to isolate individual signal paths. These techniques may include windowing and Fourier transform, principal component analysis, MUSIC and Modified MUSIC.
- the paths can be cycled through and delays can be determined.
- the algorithm can apply beamforming to all or some of the paths and determines each path's delay.
- the beamforming can be performed using the following steps:
- the LOS path is chosen.
- Delay estimates and other characteristics can be used for each of the paths to find the LOS path. Any combination of the following criteria can be used:
- the LOS delay can be output to navigation software.
- SALSA alone may not be sufficient to produce good measurements in certain environments.
- SALSA outputs measurements for first-arrival signals. These measurements may be processed by a Kalman filter; however Kalman filters are vulnerable to any biases in the input measurements. For example, if a signal from a particular GNSS satellite has no LOS component (or the LOS component is too weak to be detected) then the first arrival will correspond to a multipath component, which is necessarily delayed. This creates a bias in some measurements, and will ruin the Kalman filter performance.
- GAME is designed to identify and eliminate faulty (i.e., non-LOS) measurements.
- RAIM Receiver Autonomous Integrity Monitoring
- GAME builds on two known algorithms: Interacting Multiple Models (IMM) and Genetic Algorithms.
- IMM Interacting Multiple Models
- GAME simultaneously tracks multiple models; each model characterized by a particular allocation of LOS/non-LOS flags.
- GAME computes a Bayesian likelihood of that model being true.
- GAME assumes that multipath environment is dynamic, and accounts for a possibility of flips in LOS/non-LOS flags.
- Each epoch GAME performs two major steps computing a priori and posteriori likelihoods for a set of models. The most posteriori likely model is used to select LOS-only measurements, which are then passed to the Kalman filter for processing.
- the algorithm computes likelihoods for a set of models L n (m), where m identifies a model.
- L n (m) the probability transition matrix
- m the probability transition matrix
- the probability transition matrix can be computed from the old set to the new one. If this probability transition matrix is applied to posteriori likelihoods in the old set, we will get a priori likelihoods in the new set, which we denote ⁇ tilde over (L) ⁇ n+1 (m).
- the goal is to compute a Bayesian likelihood irrespective of the user state, which is the same as removing the condition in the above equation. This can be achieved by integrating the above equation over ⁇ right arrow over (R) ⁇ . Integration can be performed in the closed form.
- the resulting equation defines the Bayesian likelihood of a particular set of measurements with a particular set of LOS/non-LOS flags (the latter affects the covariance matrix and the normalization constant).
- This likelihood is combined with the a priori likelihood of the model to arrive at a posteriori likelihood.
- the model with the highest likelihood “wins”, and measurements identified as LOS are passed to the Kalman filter for processing.
- a starting assumption is that measurements ⁇ right arrow over (Z) ⁇ and their covariances ⁇ circumflex over (Q) ⁇ are available. Also available is the measurement matrix ⁇ .
- the unknowns include the position and clock vector ⁇ right arrow over (X) ⁇ and the set of flags ⁇ right arrow over (N) ⁇ for LOS/NLOS.
- the multipath signals are assumed to have additional bias in the measurements, and they may have a different variance as well. Additional bias is modeled as ⁇ right arrow over (G) ⁇ * ⁇ right arrow over (N) ⁇ , where ⁇ right arrow over (G) ⁇ is a vector, and ‘*’ is a per-component vector product of two vectors.
- ⁇ ⁇ ( X ⁇ , N ⁇ ) C ⁇ exp ⁇ [ - 1 2 ⁇ ( Z ⁇ - H ⁇ ⁇ X ⁇ - G ⁇ * N ⁇ ) T ⁇ Q ⁇ - 1 ⁇ ( Z ⁇ - H ⁇ ⁇ X ⁇ - G ⁇ * N ⁇ ) - 1 2 ⁇ ( X ⁇ - x ⁇ ) T ⁇ P ⁇ - 1 ⁇ ( X ⁇ - x ⁇ ) ] [ 17 ]
- Models differ by LOS/non-LOS flags for different signals, each model having its own likelihood that is set to an a priori value.
- Next likelihoods and likelihood gradients in the LOS flag space can be computed 410 .
- the current list of models is stored for future reference in memory (this list is denoted List_ 0 ).
- likelihoods for new models can be computed, each differing from a current model by flipping one LOS flag. If new models have higher likelihoods than current ones, the new models are included in the set. This is repeated some number of times.
- Models are removed from the set if their likelihoods are lower than a threshold.
- the threshold is dynamically adjusted to keep some reasonable number of models in the set.
- New models are spawned and old ones pruned to produce an updated set of models, denoted as List_ 1420 .
- the most likely model is selected and its LOS flags outputted 430 . Measurements corresponding to non-LOS signals will be discarded from processing by the navigation filter. All models are then propagated to the next epoch 440 . All models in List_ 0 and all models in List_ 1 are then looped through 450 .
- the next step is to multiply the likelihood of the model in List_ 0 by the transition probability and tally it in the likelihood of the model in List_ 1 and proceed to the next time epoch.
- One aspect the present disclosure is directed to a system and method which processes pseudorange, Doppler and ADR measurements from several navigation signal sources (such as GPS satellites), where some measurements can be substantially corrupted by the effects multipath or by other biases.
- the present disclosure processes multiple models, where each model is characterized by assuming that some signals are non-biased (e.g., not affected by multipath, or affected insignificantly), and other signals are biased (e.g., corrupted by multipath delays).
- the number of models processed can be less than the total number of possible models. For example, if there are S sources, the total possible number of models, which assume all possible allocations between biased and non-biased measurements, is equal to 2 S .
- the number of processed models can be limited by the amount of processing power available to the user.
- each processed model has a priori and posteriori likelihood values associated with it.
- the a priori likelihood computation assumes that each processed model at the current epoch has originated from a model, processed at the previous time epoch.
- the current model may be identical to one of the previous models, in which case its a priori likelihood depends on the probability of the signals not changing from biased to not-biased category and vice versa.
- the current model may differ from any model at the previous time epoch, in which case its a priori likelihood depends on the probability of a signal switching between the biased and non-biased category.
- the algorithm computes the posteriori Bayesian likelihood of measurements obtained by the receiver, This likelihood is computed by combining the a priori likelihood with the probability of obtaining the last epoch's set of measurements. It will be different for different models, since a priori probability distributions for measurement residuals are different for biased and non-biased measurements. Typically, biased measurements would have larger variances, and the expected value of residuals of biased measurements may have non-zero mean.
- the algorithm may also compute posteriori likelihoods of some derivatives of these models. For example, the algorithm may compute the likelihoods of models, where one signal source is moved from the biased to the non-biased category, or vice versa. From all candidate models (processed models and their derivatives), the algorithm selects models with highest likelihood values, which are retained. Models with lower likelihood values are destroyed. This selection process forms a set of models for computing a priori likelihoods for the next epoch. In other words, a priori likelihoods for the epoch are computed from posteriori likelihoods for the previous epoch. The model with the highest likelihood is selected. Measurements, which are flagged as non-biased for this model, are passed to the navigation filter for processing. Measurements which are deemed biased are not processed by the navigation filter.
- SoOP Signals of Opportunity
- CTG errors for Calibration, Timing, and Geolocation
- any error in the source location, reference station calibration, or reference station timing translates into an error in user TDOA ranging.
- a source is hypothetically geolocated with 50 m accuracy, the same order of magnitude accuracy for the user navigation is observed, which could severely degrade navigation accuracy.
- the errors in question are normally much smaller than other distances in the problem, e.g. those from the source to the user and from the source to reference stations.
- the ratio of the source geolocation error to the spatial scale of the problem is small, e.g. 50 m/5000 m ⁇ 10 ⁇ 2 .
- WAFFLES This method described in this disclosure is referred to as WAFFLES, which is designed to cancel effects of CTG errors in the first order approximation (with respect to the small parameter defined above). Even though this method may be an important piece of the puzzle to meet performance requirements, it should not be viewed as an excuse to relax efforts on calibrating reference stations and geolocating SoOPs accurately. Indeed, the performance of WAFFLES will depend on the geometry of the problem. Even though the first-order error is canceled, the second-order error may still be a concern for some geometries. This is when calibration and synchronization of reference stations becomes important. Moreover, the WAFFLES method allows an excellent return on any improvement in the calibration and timing. For example, reducing calibration error by a factor of 2 may reduce the second-order user ranging error by a factor of 4.
- SoOPs introduce a number of additional error sources.
- One of the most important error sources is due to timing and calibration errors at the reference station, and associated error in SoOP geolocation.
- SoOP location is not known precisely. If there is an error in the SoOP location estimate, it directly seeps into the error for TDOA and then into the user navigation error. For example, if the source is located at point 500 instead of point 510 , the time for the signal to travel 515 from the estimated position 510 to reference station 520 is greater than the time for the signal to travel 525 from the actual source position 500 to the reference station 520 which introduces a timing error. Likewise, the time for the signal to travel 535 from the estimated position 510 to user 530 is less than the time for the signal to travel 545 from the actual source position 500 to the user 530 which introduces a timing error. The timing errors will then be introduced into the TDOA measurements. Below a way to cancel effects of reference station calibration is described.
- This disclosure is directed to canceling these TDOA errors by computing a weighted sum of measurements at the cost of sacrificing two SoOP measurements.
- the approach is similar to GPS double differencing, but regular double differencing will not work here.
- double differencing cancels satellite clock error (which is a bit analogous to canceling reference station calibration errors in the present case) and satellite ephemeris errors (which are analogous to SoOP geolocation errors).
- GPS has the great advantage that a satellite is very far off, and all LOS are essentially parallel. If reference stations and the user are scattered across the area, like in our case, the plain old double differencing would not work. Hence the need for an approach like WAFFLES.
- weights can be computed for any epoch by the receiver, but they depend on the estimated user position. If the user moves around quite a bit, then weights will be time dependent. Since they are used as coefficients for terms, which define the partial derivatives of the measurement with respect to the user position, this is equivalent to changing the “virtual location” of the equivalent TOA SoOP measurement. This will not be a problem for a navigation code, which is designed from scratch properly.
- ⁇ right arrow over (x) ⁇ N ⁇ 1 ⁇ ( ⁇
- the solution for ⁇ right arrow over (x) ⁇ has two additive parts: the exact correction, and the error due to reference station calibration.
- This residual will comprise two parts: (a) The “legitimate part”, which is due to the user clock and user position error. These are unknowns we solve for, so we need that error in the residual to have something to process; and (b) The “error part”, which we do not model and do not solve for. This part is due to term ⁇ t j and due to SoOP geolocation error ⁇ right arrow over ( ⁇ ) ⁇ , which is ultimately due to ⁇ t j as well.
- U -> j S -> - R -> j ⁇ S -> - R -> j ⁇ [ 41 ] is the unit vector in the SoOP to reference station direction.
- K jm ⁇ jx a xm + ⁇ jy a ym ⁇ jm .
- ⁇ ⁇ ⁇ ⁇ s ⁇ j ⁇ ⁇ K s ⁇ ⁇ 0 ⁇ j ⁇ ⁇ ⁇ ⁇ t j [ 51 ]
- subscript s denotes different SoOPs, and not different reference stations as in (13).
- Setting ⁇ t 0 0, we combine measurements for three SoOPs with weights ⁇ s such that coefficients for ⁇ t 1 , ⁇ t 2 cancel, and the sum of weights is equal to 1 (so far).
- the WAFFLES method assumes some typical reference station position errors and some geometry (i.e., user, SoOP, and reference stations positions), computes the true and the estimated value of TDOAs for 3 SoUPs, computes coefficients ⁇ s and then combines measurements. If the algorithm works, the difference between the two weighted averages must be relatively small.
- ⁇ ⁇ 1 w ⁇ ⁇ s ⁇ ⁇ ⁇ s ⁇ ⁇ s [ 55 ]
- ⁇ s are TDOA measurements for SoOP s using reference station 0 .
- FIG. 6 illustrates a simplified block diagram of the WAFFLES method.
- Reference stations measures timing features of the transmitted signals 600 .
- the transmitted signals can be satellite signals or terrestrial signals.
- the reference stations transmit this information and source geolocation to the user via a communications channel 610 .
- Locations of the reference stations are known with a high accuracy (e.g., positions are surveyed).
- Locations of the signal transmitters is generally known approximately, with some accuracy (e.g., from design drawings for a cell tower).
- Multiple reference stations are used to geolocate each signal transmitter.
- the effects of calibration, timing, and geolocation (CTG) errors at the reference stations are mitigated by the user 620 .
- the user makes pseudorange and Doppler measurements 630 .
- Navigation equations for the user are formulated to explicitly express dependence of pseudorange on the CTG errors. Navigations equations are linearized with respect to CTG errors. Navigations equations are linearly combined to cancel the linear component of CTG-induced pseudorange errors. The linear combination measurements are output to a navigation filter 640 .
- Another aspect of the present disclosure provides a method of estimating user location under foliage or for other environments dominated by scattering of GPS signals.
- the statistical properties of the foliage are assumed to be known with some reasonable accuracy. It is not necessary to know the positions of individual twigs and leaves, rather, correlations for the channel impulse response can be used.
- the user receiver has a bank of correlators.
- the receiver can compute correlations between the replica waveform and the signal waveform at different delays.
- the channel impulse response (CIR) for a particular SoOP is a function of delay ⁇ and time t.
- This computation can be done once for each type of forest and stored at the receiver in the form of a lookup function for different pairs of angles of arrival and delays.
- arguments for this covariance matrix include time, and not the angle of arrival per se, the statistics is mostly due to the change in the angle as the time progresses and satellite SoOP moves. From a pre-stored table, the receiver can extract the covariance matrices for a particular satellite pass using the specific geometry.
- the problem can be formulated as follows. Correlations between the signal and the waveform replica can be measured. These measurements comprise a realization ⁇ tilde over (C) ⁇ of a random process. The receiver position can be estimated from this data.
- ⁇ circumflex over ( ⁇ ) ⁇ An estimate for the overall offset of the delay can be denoted with a hat, ⁇ circumflex over ( ⁇ ) ⁇ .
- ⁇ circumflex over ( ⁇ ) ⁇ ⁇ right arrow over (r) ⁇ right arrow over (l) ⁇ (t), where ⁇ right arrow over (r) ⁇ is the error in the user position estimate, and ⁇ right arrow over (l) ⁇ (t i ) is the LOS direction at time t i .
- measured correlations are in the form ⁇ tilde over (C) ⁇ ( ⁇ circumflex over ( ⁇ ) ⁇ , t).
- FIG. 7 illustrates LEAF processing.
- assumed initial pseudorange 700 pseudorange, Doppler and ADR measurements from several navigation signal sources (such as GPS satellites) are collected 710 , where measurements are corrupted by diffuse scattering of the signal, e.g. from foliage.
- Multiple correlators may be used, which estimate correlations of the signal with the signal replica. The correlators may differ by the delays of the signal replica.
- the measured correlation values are assumed to be generated by the noise and (at least for some correlators) by the direct and scattered signal.
- the present disclosure uses the statistical properties of noise and the scattered signal (which are preset in the receiver for typical environments), the present disclosure computes the most likely partition of measured values into noise and signal 720 .
- the Bayesian likelihood associated with this partition is calculated 730 .
- the algorithm maximizes the Bayesian likelihood of the noise/signal partition.
- the pseudorange value which maximizes the likelihood is the estimated pseudorange 750 .
- Embodiments of the subject matter described in this specification can be implemented as one or more computer program products, i.e., one or more modules of computer program instructions encoded on a tangible program carrier for execution by, or to control the operation of, data processing apparatus.
- the tangible program carrier can be a propagated signal or a computer readable medium.
- the propagated signal is an artificially generated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal that is generated to encode information for transmission to suitable receiver apparatus for execution by a computer.
- the computer readable medium can be a machine-readable storage device, a machine-readable storage substrate, a memory device, a composition of matter affecting a machine-readable propagated signal, or a combination of one or more of them.
- processors or processing encompasses all apparatus, devices, and machines for processing data, including by way of example a programmable processor, a computer, or multiple processors or computers.
- the processor can include, in addition to hardware, code that creates an execution environment for the computer program in question, e.g., code that constitutes processor firmware, a protocol stack, a database management system, an operating system, or a combination of one or more of them.
- a computer program (also known as a program, software, software application, script, or code) can be written in any form of programming language, including compiled or interpreted languages, or declarative or procedural languages, and it can be deployed in any form, including as a stand alone program or as a module, component, subroutine, or other unit suitable for use in a computing environment.
- a computer program does not necessarily correspond to a file in a file system.
- a program can be stored in a portion of a file that holds other programs or data (e.g., one or more scripts stored in a markup language document), in a single file dedicated to the program in question, or in multiple coordinated files (e.g., files that store one or more modules, sub programs, or portions of code).
- a computer program can be deployed to be executed on one computer or on multiple computers that are located at one site or distributed across multiple sites and interconnected by a communication network.
- the processes and logic flows described in this specification can be performed by one or more programmable processors executing one or more computer programs to perform functions by operating on input data and generating output.
- the processes and logic flows can also be performed by, and apparatus can also be implemented as, special purpose logic circuitry, e.g., an FPGA (field programmable gate array) or an ASIC (application specific integrated circuit).
- processors suitable for the execution of a computer program include, by way of example, both general and special purpose microprocessors, and any one or more processors of any kind of digital computer.
- a processor will receive instructions and data from a read only memory or a random access memory or both.
- the essential elements of a computer are a processor for performing instructions and one or more memory devices for storing instructions and data.
- a computer will also include, or be operatively coupled to receive data from or transfer data to, or both, one or more mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks.
- mass storage devices for storing data, e.g., magnetic, magneto optical disks, or optical disks.
- a computer need not have such devices.
- a computer can be embedded in another device, e.g., a mobile telephone, a personal digital assistant (PDA), a mobile audio or video player, a game console, a Global Positioning System (GPS) receiver, to name just a few.
- PDA personal digital assistant
- GPS Global Positioning System
- Computer readable media suitable for storing computer program instructions and data include all forms of non volatile memory, media and memory devices, including by way of example semiconductor memory devices, e.g., EPROM, EEPROM, and flash memory devices; magnetic disks, e.g., internal hard disks or removable disks; magneto optical disks; and CD ROM and DVD-ROM disks.
- semiconductor memory devices e.g., EPROM, EEPROM, and flash memory devices
- magnetic disks e.g., internal hard disks or removable disks
- magneto optical disks e.g., CD ROM and DVD-ROM disks.
- the processor and the memory can be supplemented by, or incorporated in, special purpose logic circuitry.
- embodiments of the subject matter described in this specification can be implemented on a computer having a display device, e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor, for displaying information to the user and a keyboard and a pointing device, e.g., a mouse or a trackball, by which the user can provide input to the computer.
- a display device e.g., a CRT (cathode ray tube) or LCD (liquid crystal display) monitor
- keyboard and a pointing device e.g., a mouse or a trackball
- Other kinds of devices can be used to provide for interaction with a user as well; for example, input from the user can be received in any form, including acoustic, speech, or tactile input.
- Embodiments of the subject matter described in this specification can be implemented in a computing system that includes a back end component, e.g., as a data server, or that includes a middleware component, e.g., an application server, or that includes a front end component, e.g., a client computer having a graphical user interface or a Web browser through which a user can interact with an implementation of the subject matter described is this specification, or any combination of one or more such back end, middleware, or front end components.
- the components of the system can be interconnected by any form or medium of digital data communication, e.g., a communication network. Examples of communication networks include a local area network (“LAN”) and a wide area network (“WAN”), e.g., the Internet.
- LAN local area network
- WAN wide area network
- the computing system can include clients and servers.
- a client and server are generally remote from each other and typically interact through a communication network.
- the relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other.
Abstract
Description
where m is the number of elements in the array, and n is the rank of the signal subspace (i.e., number of spectral components we are looking for). Coefficients c1 are unknowns and will be solved for.
C*S=0. [2]
φc=μ [3]
where the (m−n)n×n matrix φ and the (m−n)n×1 vector μ are entirely determined from the elements of S, and where c=[c1, . . . , cn]. If the sample version of S is used, then c can be treated as unknowns in a linear (over-determined) system of equations and can be solved for.
A(z)=1+c 1 z −1 + . . . +c n z −n. [4]
Although it may appear convoluted, it is recognized that this method works for correlated and for coherent signals, i.e. assumption of signal independence is not necessary.
M ij =E{y i ·y j*}. [6]
Signal samples yi are a sum of multiple signal components. In the case of fully coherent signal components, all components (except noise) vary by the same phase with time, and therefore
E{y i ·y j *}=y i ·y j*. [7]
M{right arrow over (x)}=g{right arrow over (y)} [8]
where g is a constant. Thus, matrix M projects any vector on vector {right arrow over (y)}, which means that M has only one non-zero eigenvector, which is {right arrow over (y)}. (If there is noise, there will be noise space eigenvectors, but the present disclosure is concerned with the signal space.)
where notation c0=1 is used for brevity. This assumes that samples yl comprise n signal components arriving from different directions. If there is a uniform linear array (ULA), then
Changing the order of summation to get:
Equations [12] for different l form a linear system with linearly independent coefficients akeiω
for any value of k.
Denoting eiω
where bk are defined via roots of polynomial [13]. Equation [14] must hold for any k, which means that bk are simply ζk −1 (not necessarily in the same order).
-
- 1. Get samples yl. Average them over time to decrease noise.
- 2. Form equations
and solve them for cq. Assume c0=1.
-
- 3. Find roots of the polynomial, which has cq for coefficients.
- 4. Roots will be equal to e−iω
k where ωk defines a direction of arrival for the k-th signal components in a ULA.
-
- (1) each correlator is placed at a different delay;
- (2) the correlator delays are computed using the estimated user position, user clock error, satellite clock error and satellite position (this is known as ultra-tight coupling in integrated GPS-INS systems), and account for the LOS propagation;
- (3) the delays are set in such a way that the expected LOS signal has non-zero correlations with at least some replicas; and
- (4) the correlations are estimated at regular time intervals (sub-epochs).
-
- (1) one dimension is the sub-epochs, and the other dimension is delay; and
- (2) the 2D matrix represents measurements at a single epoch.
-
- (1) the phases of correlation measurements are rotated by an amount consistent with the user motion (the phase of correlation for the LOS signal will be static across sub-epochs after this correction); and
- (2) the correlation matrix is interpolated in the delay domain to compensate for changes of the delay from one epoch to another;
-
- (1) LOS is the earliest arriving path;
- (2) LOS is the slowest varying principal component; and
- (3) LOS corresponds to a nearly zero phase rate in the Fourier domain.
-
- a. For the current path in the loop, it treats that path as a signal and all other paths as interference. It computes beamforming weights, which maximize the gain in the direction of the current path, while placing nulls in the directions of the other N−1 paths. These beamforming weights are applied to data from all frequency channels.
- b. The result of the previous step is M complex numbers, which are in the channel domain. Modified MUSIC is applied again, this time in the channel domain. Fewer paths are now sought for two reasons. First, time domain/DOA analysis has separated many paths already, and for this particular DOA there should be one or at most few paths left, which were not resolved. Second, having only M measurements, we would not be able to find many paths anyway. The algorithm currently looks for K paths.
- c. For each of the K paths, the algorithm computes beamforming weights, places max gain on the current path and null on K−1 other paths.
-
- a. Spectral frequency in the time domain. This value is mapped to the delta range measurement residual for that path. (Or to DOA, if desired.)
- b. Spectral frequency in the channel domain. This value is mapped to the delay measurement residual for that path.
- c. Amplitude of the signal on that path, as computed by the beamforming. Note that beamforming is applied twice (in the DOA domain and then in the delay domain.)
-
- a. Delay residual is not too large (within a configurable threshold)
- b. Delta range residual is not too large (within a configurable threshold)
- c. First arrival among those satisfying criteria a and b above.
-
- a. Compute beamforming weights to put gain on the desired path (i.e., DOA) and/or nulls on other paths' DOA
- b. Apply weights to multiple correlators, which have different replica delays or track different frequency channels of the same signal
- c. Determine delay using any known method, such as early-minus-late correlators, MEDLL, strobe correlator, or super-resolution in the delay or channel domain.
-
- a. TOA of the LOS path is the earliest for all paths
- b. Post-beamforming signal amplitude is above some threshold level. The threshold may be determined by the SNR and/or noise level
- c. DOA of the LOS path is within some tolerance from the estimated direction to the signal source.
-
- 1. GAME is able to identify multiple faults,
- 2. GAME makes use of history of measurements, and
- 3. GAME yields a quantitative measure of the integrity of measurements in the form of measurement likelihood.
where {circumflex over (Q)} is the covariance matrix {hacek over ({right arrow over (x)}({right arrow over (R)}) are expected values of measurements for user state {right arrow over (R)}, and {right arrow over (x)} are measurements. This probability density function is conditioned on the user state, which is unknown. The goal is to compute a Bayesian likelihood irrespective of the user state, which is the same as removing the condition in the above equation. This can be achieved by integrating the above equation over {right arrow over (R)}. Integration can be performed in the closed form. The resulting equation defines the Bayesian likelihood of a particular set of measurements with a particular set of LOS/non-LOS flags (the latter affects the covariance matrix and the normalization constant). This likelihood is combined with the a priori likelihood of the model to arrive at a posteriori likelihood. As was stated above, the model with the highest likelihood “wins”, and measurements identified as LOS are passed to the Kalman filter for processing.
E=({right arrow over (Z)}−Ĥ{right arrow over (X)}−{right arrow over (G)}*{right arrow over (N)})T·{circumflex over (Q)}−1·({right arrow over (Z)}−Ĥ{right arrow over (X)}−{right arrow over (G)}*{right arrow over (N)})+({right arrow over (X)}−{right arrow over (x)})T ·{circumflex over (P)} −1·({right arrow over (X)}−{right arrow over (x)})={right arrow over (X)} T·(Ĥ T ·{circumflex over (Q)} −1 ·Ĥ+{circumflex over (P)} −1)·{right arrow over (X)}+{right arrow over (X)} T ·Ĥ T ·{circumflex over (Q)} −1·{right arrow over (ξ)}+{right arrow over (ξ)}T ·{circumflex over (Q)} −1 ·Ĥ·{right arrow over (X)}+{right arrow over (ξ)} T ·{circumflex over (Q)} −1 ·{right arrow over (ξ)}+{right arrow over (X)} T ·{circumflex over (P)} −1 ·{right arrow over (X)}−{right arrow over (X)} T ·{circumflex over (P)} −1 ·{right arrow over (x)}−{right arrow over (x)} T ·{circumflex over (P)} −1 ·{right arrow over (X)}+{right arrow over (x)} T ·{right arrow over (P)} −1 ·{right arrow over (x)} [18]
where {right arrow over (ξ)}={right arrow over (G)}*{right arrow over (N)}−{right arrow over (Z)} is denoted, for brevity.
{right arrow over (X)}={circumflex over (M)}·{right arrow over (ρ)} [19]
{right arrow over (X)} T·(Ĥ T ·{circumflex over (Q)} −1 ·Ĥ+{circumflex over (P)} −1)·{circumflex over (X)}={right arrow over (ρ)} T ·{circumflex over (M)} T·(Ĥ T ·{circumflex over (Q)} −1 ·Ĥ+{circumflex over (P)} −1)·{circumflex over (M)}·{right arrow over (ρ)} [20]
{circumflex over (M)}=Ĵ T
where Ĵ is a factor in the LU-decomposition
Ĥ T ·{circumflex over (Q)} −1 ·Ĥ+Ĥ{circumflex over (P)} −1 =Ĵ·Ĵ T. [22]
In the new variables, the argument of the exponent takes the following form:
E={circumflex over (ρ)} T·{circumflex over (ρ)}+2·{circumflex over (D)} T·{right arrow over (ρ)}+F=({right arrow over (ρ)}+{right arrow over (D)})2 +F−D 2 [23]
where the following notations are used:
{right arrow over (D)}={circumflex over (M)} T·(Ĥ T ·{circumflex over (Q)} −1 ·{circumflex over (ξ)}−{circumflex over (P)} −1 ·{circumflex over (x)})F={right arrow over (ξ)} T ·{circumflex over (Q)} −1 ·{right arrow over (ξ)}+{right arrow over (x)} T ·{circumflex over (P)} −1 ·{right arrow over (x)} [28]
Substituting the expression for E into that for Λ({right arrow over (X)},{right arrow over (N)}) and integrate the result with respect to {right arrow over (X)}:
The constant multiplier and ∥{circumflex over (M)}∥ will be canceled when likelihood values for all models are normalized, and therefore do not have to be computed. This is the final (albeit still not normalized) equation for the likelihood of each model. It accounts for both the new measurements and for the prior information in the form of a navigation solution.
δ{right arrow over (ρ)}=K·Δ{right arrow over (t)} [26]
This is the un-modeled error for the user for TDOA measurements using different reference stations. The error is due to reference station calibration errors AI and SoOP geolocation errors (which are in turn due to reference station calibration errors). If there are N reference stations, there are N components in the calibration error Δ{right arrow over (t)} and correspondingly N TDOA measurements (each with respect to a different reference station) per SoOP.
ρj =|{right arrow over (R)} j −{right arrow over (S)}|+θ+Δt j [27]
where {right arrow over (R)}j is the position of the reference station, {right arrow over (S)} is position of the source, θ is the true timing of some feature in the SoOP signal, and Δtj is the calibration error.
{right arrow over (Ŝ)}={right arrow over (S)} 0 +δ{right arrow over (S)}. [28]
The unknowns are δ{right arrow over (S)}, θ, which can be combined into one vector δx={δ{right arrow over (S)},θ}. Then in the first order the solution for δx is obtained from a linear system of equations:
δ{right arrow over (x)}=N −1·(ρ−|{right arrow over (R)}−{right arrow over (S)} 0 |−Δt) [29]
where components of 4-vector ρ−|{right arrow over (R)}−{right arrow over (S)}0|−Δt correspond to different reference stations (i.e., j-th component is ρj−|{right arrow over (R)}j−{right arrow over (S)}0|−Δtj) and where the 4×4 matrix N is defined by the following:
The solution for δ{right arrow over (x)} has two additive parts: the exact correction, and the error due to reference station calibration. They can be separated:
δ{right arrow over (x)}=δ{circumflex over (x)} 0+{right arrow over (ε)} [31]
where
δ{right arrow over (x)} 0 =N −1·(ρ−|{right arrow over (R)}−{right arrow over (S)} 0|)
{right arrow over (ε)}=N −1 ·Δt [32]
Vector {right arrow over (ε)} is the SoOP geolocation error. As we shall see shortly, it affects the user position error.
ρu =|{right arrow over (Y)}−{right arrow over (S)}|+θΔt u [33]
where {right arrow over (Y)} is the true user position, and Δtu is the user clock error. The true TDOA with the reference station pseudorange is as follows:
ρu−ρj =|{right arrow over (Y)}−{right arrow over (S)}|+Δt u −|{right arrow over (R)} j −{right arrow over (S)}|−Δt j. [34]
To process this measurement, it is compared with the estimate for TDOA, i.e. compute the residual. This residual will comprise two parts: (a) The “legitimate part”, which is due to the user clock and user position error. These are unknowns we solve for, so we need that error in the residual to have something to process; and (b) The “error part”, which we do not model and do not solve for. This part is due to term Δtj and due to SoOP geolocation error {right arrow over (ε)}, which is ultimately due to Δtj as well.
δ{right arrow over (ρ)}=M{right arrow over (ε)}−Δ{right arrow over (t)} [35]
where 4×4 matrix M is defined by:
Substituting {right arrow over (ε)} gets:
δ{right arrow over (ρ)}=(MN −1 −I)·Δ{right arrow over (t)}. [37]
Next, we introduce the notation
K=MN −1 −I [38]
to get:
δ{right arrow over (ρ)}=K·Δ{right arrow over (t)} [39]
δ{right arrow over (S)}·{right arrow over (U)} j =Δ{right arrow over (t)} j+δθ [40]
where
is the unit vector in the SoOP to reference station direction.
where subscript k is for x, y components and
and
U pqk =U pk −U qk
D=U 01y U 02x −U 02y U 01x [44]
User TDOA error is given by (see (8)):
δρj=ξjx S x+ξjy S y +Δt j [45]
where
Thus, matrix K is given by:
K jm=ξjx a xm+ξjy a ymδjm. [47]
Next we compute Kjm explicitly for some specific m, for example for m=0 and for different values of j.
Case m=0; j=
Case m=0; j=2
Case in =0; j=0
Amazingly, K00=K10=K20. The same holds true for other values of m.
where subscript s denotes different SoOPs, and not different reference stations as in (13). Setting Δt0=0, we combine measurements for three SoOPs with weights αs such that coefficients for Δt1, Δt2 cancel, and the sum of weights is equal to 1 (so far). This produces the following system of equations for as:
K 001α0 +K 101α1 +K 201α2=0
K 002α0 +K 102α1 +K 202α2=0
α0+α1+α2=1
If TDOA measurements from 3 SoOPs are linearly combined with weights αs, the first order error cancels.
-
- 1. Geolocation is done in 2D. Vertical positions of SoOPs are assumed to be known with some accuracy. The truth measurements and the estimated measurements are simulated using the true and the estimated vertical positions of SoOPs respectively.
- 2. User position is not known perfectly either. If the user were located exactly at the estimated position, the net results of using TDOA measurements with WAFFLES algorithm would be the measurement residual in the amount of the error, not compensated by WAFFLES. If the user is off its estimated position (i.e., if there is a position error estimate by the navigation filter), then there is an additional error source. This error is due to an error in the LOS direction to the SoOP (LOS is computed the estimated SoOP position instead of the true one). The magnitude of this error is the difference of the projection of the user position error on the estimated LOS and the true LOS. Assuming that the vector of the user position error can point at any direction equally probable means that this additional error can be negative, positive, or zero. The average magnitude of it is 1/√{square root over (2)} of the maximum error. The same is true for the SoOP geolocation error. Two factors of 1/√{right arrow over (2)} produce a factor of ½, yielding the following:
-
- where δ{right arrow over (U)} is the user position error, δ{right arrow over (S)} is the SoOP geolocation error, and {right arrow over (U)}−{right arrow over (S)} is the range to the SoOP.
There is just one more little modification to the equations to be made. The weighted sum of TDOA measurements can be used in the Kalman filter for navigating the user. Normally, a TOA or TDOA measurement equation has the position of the user in the form like |{right arrow over (U)}−{right arrow over (S)}|+const. This means that partial derivatives of the measurement equation with respect to the user position form a unit vector in the direction of the source. Thus, the RSS (root sum squared) of the partials is equal to one. Note that α0+α1+α2=1 for weighted TDOA does not guarantee this condition. In order to imitate this condition for the new weighted TDOA measurement, it should be scaled. (In general, this is not necessary, but keeps the lineage to the conventional TOA/TDOA processing; scaling becomes necessary if the Kalman filter formulation is not changed from the conventional TOA/TDOA processing.) Thus, partial derivatives can be computed with respect to the user position and then the scaling can be computed:
- where δ{right arrow over (U)} is the user position error, δ{right arrow over (S)} is the SoOP geolocation error, and {right arrow over (U)}−{right arrow over (S)} is the range to the SoOP.
Now the desired weighted TDOA measurement is given by:
where ρs are TDOA measurements for SoOP s using
R=R(τ,t) [56]
The CIR has the LOS peak at τ=0, which corresponds to the true user position.
C(τ,t)=R(τ,t)*A(τ)=∫R(θ,t)·A(τ−θ)dθ. [57]
Assuming that statistical properties of R(τ, t) are known, pair-wise covariances for values of C(τ,t) can be computed:
Q(ρl ,t lτj ,t j)=E{C(τi ,t i)·C*(τj ,t j)}=∫∫R(θl ,t l)·A(τ1−θl)R*(θ2 ,t j)·A(τj−θ2)dθ 1 dθ 2 [58]
Similarly, a noise covariance matrix can be computed
η(τi ,t i,τj ,t j)=E{n(τi ,t i)·n*(τj ,t j)}. [59]
This computation can be done once for each type of forest and stored at the receiver in the form of a lookup function for different pairs of angles of arrival and delays. (Even though arguments for this covariance matrix include time, and not the angle of arrival per se, the statistics is mostly due to the change in the angle as the time progresses and satellite SoOP moves. From a pre-stored table, the receiver can extract the covariance matrices for a particular satellite pass using the specific geometry.)
{tilde over (C)}({circumflex over (τ)},t)=C(τ,t)+n(τ,t) [60]
is the most likely realization of the random process for C(τ,t) and n(τ,t).
Mx={tilde over (C)}({circumflex over (τ)},t) [61]
where matrix M has the following structure:
F=xWx* [63]
where the covariance matrix W is constructed from the covariance matrices for the signal and for the noise:
Claims (16)
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